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Keywords:
Wold-type decomposition; quasi-isometry; lifting
Wold-type decomposition; quasi-isometry; lifting
Summary:
This paper investigates the necessary and sufficient conditions under which a quasi-isometry $T$ on a Hilbert space ${\mathcal H}$ admits a Wold-type decomposition in Shimorin's sense. We establish a close connection between this decomposition and the kernel condition $T^*T {\mathcal N} (T^*)\subset {\mathcal N} (T^*)$, where ${\mathcal N}(T^*)$ is the kernel of the adjoint operator $T^*$ of $T$. Additionally, we discuss conditions related to certain cyclic and wandering subspaces, as well as the role of the Cauchy dual operator of $T$. Furthermore, we examine operators similar to contractions, that admit quasi-isometric liftings satisfying the kernel condition. This analysis leads to the identification of a special class of quasicontractions with such liftings, and on the other hand, to the construction of certain expansive quasi-isometric liftings $S_{\alpha }$ ($0<\alpha <1)$.
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